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The Risk revised edition rules have a Major Reward that grants the defender a bonus defense die. This die will replace the lowest defender's die rolled (unless the bonus die is the lowest).

What are the expected troop losses for the Attackers and Defenders when the Attacker attacks with 3 troops, and you choose to roll a single Defense die and your bonus Defense die versus when you roll two Defense dice and your bonus Defense die?

I have worked out the math for the single Defense die plus bonus Defense die. This is the easiest calculation, because we only care about the highest die rolled for the attacker (out of 216 possibilites), and the highest die rolled for the defender (out of 36 possibilities).

In the 3 Attacker dice versus the 2 Defender dice plus 1 bonus dice, both players throw away their lowest die. So, all that is needed to calculate the losses is to calculate the chances of each two highest die rolls from three dice (out of 216), and then figure out the losses (although, the results are a little more complicated because you have a Win/Loss/Tie (each lose 1).